26
nections while smaller, peripheral airports the
number of which is higher, have only a few connections with other airports. In the language of
mathematics this means that the degree distribution in this latter network (the distribution of the
number of points having k connections) is according to a power function. This network is called
scale-free
1
network characterised by a few large
centre points with very high number of connections therefore these are dominant in the structure of the network.
Power-function distribution is typical for the
most of complex networks, i.e. the majority of
natural networks are scale-free. Their centre
points significantly influence or often determine
the stability, dynamic behaviour of a system and
its resistance against faults and attacks.
Scientists noted that real networks are not random. Power functions have an important role in
the fields of chaos theory, fractals and phase
changes. According to Barabási (2003) complex
networks (complex systems, chaotic networks)
are also based on law.
An important feature of complex systems is
the capability of self-organisation. Power functions are apparent signs of self-organisation and
order centre points observable in networks are
the result of power functions. Why centre points
appear in every network? Barabási gives the following reply to the question. Most real networks
have something in common: growth. This on its
own, however, is not enough to answer the question. Therefore in the course of studying webpages, the significant role of popularity has been
shown in the process of increasing number of
connections: when we decide where to link on
the World Wide Web connection is made on the
basis of popularity. Real networks are thus led by
two laws: growth and popular links.
In the course of network growth, older points
have more time to establish connections than
more recent ones therefore old points will be the
1 In random networks the degree number of points has a
typical size, a scale which is determined by the top (typical point) of the degree distribution graph (bell curve in
Fig. 2.8). In the case of power function, the distribution
has now top, neither a typical scale nor typical points.
richest in connections. Furthermore, such points
will be chosen more frequently as a result of the
high number of their existing connections therefore they will have a lot of links: they become
centre points. This is the “rich get richer”
phenomenon.
Growth and popularity connections could
explain the basic characteristics of networks
existing in nature, like, for example, the network
of metabolism within a cell. Scale-free topology
became important for many scientific branches. It
also brought further questions. One of the most
significant ones is that how elements connected
to the system later get along.
There are different answers to the above question according to the two types of networks.
In one network type scale-free topology
remains despite competition among the elements.
In the other network type the principle of “the
winner takes it all” prevails. The latter network is
not scale-free, it has one large centre instead and
numerous smaller points exist beside it the links
of which are oriented towards the centre forming
a star-shaped network. This is called suitability
model, i.e. the principle “the suitable gets richer”
succeeds in the operation and growth of the network. This model is practically suitable for simulating the growth of the capitalist economy: it
describes the network in competition.
An important issue is the fault tolerance of
networks (systems). Studies proved that life, living systems are much more resistant to errors
than artificial systems created by humans. Some
of the latter ones could be disabled even by the
failure of a single component.
Natural systems increase their fault tolerance
by forming networks with multiple interconnections. The detailed study of the Internet also
proved that it resembles the network of living
systems in many ways forming a network of high
fault tolerance. Barabási (2003) formulated its
mathematical background as well.
The scale-free degree is either smaller than three or
equals it. Surprisingly the majority of networks
interesting to us – from the internet to cells – are
scale-free and their degree is smaller than three.
Therefore these networks fall apart only if all
points are removed, i.e. practically never. (Barabási
2003, p. 158).
2 Structure and Operation of Systems, Models of the Global Earth System
nections while smaller, peripheral airports the
number of which is higher, have only a few connections with other airports. In the language of
mathematics this means that the degree distribution in this latter network (the distribution of the
number of points having k connections) is according to a power function. This network is called
scale-free
1
network characterised by a few large
centre points with very high number of connections therefore these are dominant in the structure of the network.
Power-function distribution is typical for the
most of complex networks, i.e. the majority of
natural networks are scale-free. Their centre
points significantly influence or often determine
the stability, dynamic behaviour of a system and
its resistance against faults and attacks.
Scientists noted that real networks are not random. Power functions have an important role in
the fields of chaos theory, fractals and phase
changes. According to Barabási (2003) complex
networks (complex systems, chaotic networks)
are also based on law.
An important feature of complex systems is
the capability of self-organisation. Power functions are apparent signs of self-organisation and
order centre points observable in networks are
the result of power functions. Why centre points
appear in every network? Barabási gives the following reply to the question. Most real networks
have something in common: growth. This on its
own, however, is not enough to answer the question. Therefore in the course of studying webpages, the significant role of popularity has been
shown in the process of increasing number of
connections: when we decide where to link on
the World Wide Web connection is made on the
basis of popularity. Real networks are thus led by
two laws: growth and popular links.
In the course of network growth, older points
have more time to establish connections than
more recent ones therefore old points will be the
1 In random networks the degree number of points has a
typical size, a scale which is determined by the top (typical point) of the degree distribution graph (bell curve in
Fig. 2.8). In the case of power function, the distribution
has now top, neither a typical scale nor typical points.
richest in connections. Furthermore, such points
will be chosen more frequently as a result of the
high number of their existing connections therefore they will have a lot of links: they become
centre points. This is the “rich get richer”
phenomenon.
Growth and popularity connections could
explain the basic characteristics of networks
existing in nature, like, for example, the network
of metabolism within a cell. Scale-free topology
became important for many scientific branches. It
also brought further questions. One of the most
significant ones is that how elements connected
to the system later get along.
There are different answers to the above question according to the two types of networks.
In one network type scale-free topology
remains despite competition among the elements.
In the other network type the principle of “the
winner takes it all” prevails. The latter network is
not scale-free, it has one large centre instead and
numerous smaller points exist beside it the links
of which are oriented towards the centre forming
a star-shaped network. This is called suitability
model, i.e. the principle “the suitable gets richer”
succeeds in the operation and growth of the network. This model is practically suitable for simulating the growth of the capitalist economy: it
describes the network in competition.
An important issue is the fault tolerance of
networks (systems). Studies proved that life, living systems are much more resistant to errors
than artificial systems created by humans. Some
of the latter ones could be disabled even by the
failure of a single component.
Natural systems increase their fault tolerance
by forming networks with multiple interconnections. The detailed study of the Internet also
proved that it resembles the network of living
systems in many ways forming a network of high
fault tolerance. Barabási (2003) formulated its
mathematical background as well.
The scale-free degree is either smaller than three or
equals it. Surprisingly the majority of networks
interesting to us – from the internet to cells – are
scale-free and their degree is smaller than three.
Therefore these networks fall apart only if all
points are removed, i.e. practically never. (Barabási
2003, p. 158).
2 Structure and Operation of Systems, Models of the Global Earth System